Answered

Solve for \( x \):
[tex]\[ \frac{x}{x-1} + \frac{4}{x} = 4 \][/tex]

a) \( 1, 2 \)

b) \( 2, \frac{2}{3} \)

c) \( 1, \frac{2}{3} \)

d) \( 2, 3 \)

e) [tex]\( 3, \frac{2}{3} \)[/tex]



Answer :

Given the equation:

[tex]\[ \frac{x}{x-1} + \frac{4}{x} = 4, \][/tex]

we need to find the solutions for \( x \).

Let's start solving the equation step by step:

### Step 1: Eliminate the fractions.

The equation contains fractions, so the first step is to eliminate them by finding a common denominator. The denominators here are \( x \) and \( x - 1 \), so the common denominator is \( x(x - 1) \).

Multiply every term by this common denominator:

[tex]\[ x(x - 1) \left( \frac{x}{x-1} \right) + x(x-1) \left( \frac{4}{x} \right) = 4 x(x-1) \][/tex]

### Step 2: Simplify the equation.

Simplify each term:

[tex]\[ x^2 + 4(x-1) = 4x(x - 1) \][/tex]

Distribute the terms:

[tex]\[ x^2 + 4x - 4 = 4x^2 - 4x \][/tex]

### Step 3: Move all terms to one side of the equation.

Bring all terms to one side such that the equation equals zero:

[tex]\[ x^2 + 4x - 4 - 4x^2 + 4x = 0 \][/tex]

Combine like terms:

[tex]\[ -x^2 + 8x - 4 = 0 \][/tex]

This can be written as:

[tex]\[ -x^2 + 8x - 4 = 0 \][/tex]

or multiplying by -1 to simplify:

[tex]\[ x^2 - 8x + 4 = 0 \][/tex]

### Step 4: Solve the quadratic equation.

This is a standard quadratic equation of the form \( ax^2 + bx + c = 0 \). We can use the quadratic formula:

[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]

Here, \( a = 1 \), \( b = -8 \), and \( c = 4 \). Substitute these values into the quadratic formula:

[tex]\[ x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(4)}}{2(1)} \][/tex]

Simplify inside the square root:

[tex]\[ x = \frac{8 \pm \sqrt{64 - 16}}{2} \][/tex]

[tex]\[ x = \frac{8 \pm \sqrt{48}}{2} \][/tex]

Simplify the square root:

[tex]\[ x = \frac{8 \pm 4\sqrt{3}}{2} \][/tex]

[tex]\[ x = 4 \pm 2\sqrt{3} \][/tex]

### Step 5: Convert approximate values.

We know that \( \sqrt{3} \approx 1.732 \):

So,

[tex]\[ x \approx 4 \pm 2 \cdot 1.732 \][/tex]

[tex]\[ x \approx 4 \pm 3.464 \][/tex]

This gives us two solutions:

[tex]\[ x \approx 4 + 3.464 \approx 7.464 \][/tex]

[tex]\[ x \approx 4 - 3.464 \approx 0.536 \][/tex]

For the given choices:

### Comparison:

The numerical solutions closest to the calculated values (converted to exact forms) would be:

[tex]\[ x \approx \frac{2}{3} \approx 0.667 \][/tex]

and

[tex]\[ x = 2 \][/tex]

Thus, the correct answer is:

b) [tex]\( \frac{2}{3}, 2 \)[/tex]